A cloud of smoke hung over the camp as the Lebanese army continued to shell Fatah al-Islam positions.
The fighters returned fire with mortar bombs and rocket-propelled grenades.
The Red Cross employees are the first aid workers to be killed in the conflict.
Virginian de la Guardia, of the Red Cross in Lebanon, said that it has become increasingly difficult to get into the camp.
"The Palestinian refugees are not being treated properly by Lebanon"
Sunny, Ottawa, Canada
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"It has been very difficult to get assistance to the people who are still trapped inside, and also difficult for those who want to leave. Assistance in terms of food and water has also been very minimal."
Medics told the AFP news agency the scholar was from the Palestinian Scholars' League and said he had been lightly wounded in the leg in the incident.
"I'm alright. The (peace) initiative will continue ... to stop the bloodletting," Hajj said.
On Sunday relief workers evacuated 250 people from the camp, where more than 3,000 inhabitants out of an original 31,000 remain trapped. The rest have fled.
Fighting erupted in the camp on May 20 and has killed at least 60 Lebanese soldiers, 50 fighters and 20 civilians.
Lebanese authorities say the fighting was first sparked by raids on Fatah al-Islam hideouts in the nearby port city of Tripoli following a bank robbery.
However, Fatah al-Islam says it acted in self-defence and has vowed to fight to the death.
The fighting has further undermined Lebanon's stability at a time when the country is struggling to resolve a seven-month-old political crisis over power-sharing and proposals for an international tribunal to try the alleged killers of Rafik Hariri, the assassinated former prime minister.